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A Computational Investigation of the Geometrical and Electronic Structures of VGe (n = 1-4) Clusters by Density Functional Theory and Multiconfigurational CASSCF/CASPT2 Method n-/0

Van Tan Tran, Minh Thao Nguyen, and Quoc Tri Tran J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b08351 • Publication Date (Web): 20 Sep 2017 Downloaded from http://pubs.acs.org on September 20, 2017

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The Journal of Physical Chemistry

A Computational Investigation of the Geometrical and Electronic Structures of VGen−/0 (n = 1-4) Clusters by Density Functional Theory and Multiconfigurational CASSCF/CASPT2 Method

Van Tan Tran(a)(*), Minh Thao Nguyen(a)(b), Quoc Tri Tran(a) (a) Theoretical and Physical Chemistry Division, Dong Thap University, Cao Lanh City, Dong Thap, 870200, Vietnam (b) Department of Chemistry, University of Science, Vietnam National University - Ho Chi Minh City, Ho Chi Minh City, 748000, Vietnam

Abstract Density functional theory and multiconfigurational CASSCF/CASPT2 method have been employed to study the low-lying states of VGen−/0 (n = 1 - 4) clusters. For VGe−/0 and VGe2−/0 clusters, the relative energies and geometrical structures of the low-lying states are reported at the CASSCF/CASPT2 level. For the VGe3−/0 and VGe4−/0 clusters, the computational results show that due to the large contribution of the Hartree-Fock exact exchange, the hybrid B3LYP, B3PW91, and PBE0 functionals overestimate the energies of the high-spin states as compared to the pure GGA BP86 and PBE functionals and the CASPT2 method. Based on the pure GGA BP86 and PBE functionals and the CASSCF/CASPT2 results, the ground states of anionic and neutral clusters are defined, the relative energies of the excited states are computed, and the electron detachment energies of the anionic clusters are evaluated. The computational results are employed to give new assignments for all features in the photoelectron spectra of VGe3− and VGe4− clusters.

(*) Corresponding author:

Van Tan Tran Email: [email protected] Telephone:(+84)1228942399 1 ACS Paragon Plus Environment


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I. Introduction Germanium clusters attract great attentions of scientists because of their important applications in the semiconductor industry and synthesis of novel materials. However, the pure germanium clusters are chemically reactive and therefore not suitable as a building block of clusterassembled materials. In searching for the stable clusters that could be used as building blocks, appropriate transition metal atoms are doped on the pure germanium clusters. The doping of transition metal strongly affects the structure, stability, and electronic and magnetic properties of germanium clusters. In the literature, we can see a large amount of experimental and theoretical investigations on the transition metal doped germanium clusters.1-11 These studies show that the doping of transition metal can stabilize the Ge cage structures and the stable transition metal doped germanium clusters can be used as building blocks for cluster-assembled materials.1,2,9-11 In this work, we are interested in the small-sized vanadium doped germanium clusters. The geometrical and electronic structures of VGen− (n = 3 - 12) clusters were probed by photoelectron spectroscopy in which the electron detachment energies of the anionic clusters were reported.2 In the same work, density functional theory with the B3PW91 functional was employed to investigate the geometrical and electronic structures of VGen−/0 (n = 3 - 12) clusters and to interpret the photoelectron spectra of anionic clusters.2 Also, the structure, stability, and electronic and magnetic properties of VGen (n = 1 - 19) clusters were investigated with the PBE functional.1 The vertical electron affinities of the VGen (n = 3 - 12) clusters as computed with the PBE functional were found to be 0.4 - 0.8 eV smaller than the vertical electron detachment energies of the anionic clusters as calculated with the B3PW91 functional.1,2

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From the literature, we can see that density functional theory is widely used to study vanadium doped germanium clusters.1,2,12 However, because of a single-reference method, density functional theory cannot access all the excited states which are important to explain all features in the photoelectron spectra of anionic clusters. That is the reason why only the lowest band in the photoelectron spectra of VGen− (n = 3 - 12) clusters are interpreted based on the density functional theory results, while the remaining bands are not explained.2 Moreover, because of the near degeneracy of the low-lying electronic states as can be seen in the case of transition metal containing compounds, the relative energies of the low-lying states and the relative stabilities of the isomers predicted by different functionals can be different. This feature can be seen in the case of VSi3− cluster where the B3LYP functional predicts the 5B2 state of rhombic isomer as the anionic ground state, while the BP86 functional favors the 1A′ of tetrahedral isomer.13 For the VGe3 cluster, the B3PW91 functional calculates the rhombic isomer in C2v symmetry as the most stable isomer, while the tetrahedral and cyclic isomers are 0.001 and 0.48 eV less stable.2 In contrast, the PBE functional predicts the tetrahedral isomer in C3v symmetry as the most stable isomer, while the cyclic isomer in C2v symmetry is 0.02 eV less stable.1 In such cases, the multiconfigurational CASSCF/CASPT2 method needs to be employed. With the ability to calculate the energies of all the important ground and excited states with high accuracy, the CASSCF/CASPT2 have been known as a sufficient method to investigate the low-lying states of transition metal containing compounds.13-16 Also, the electron detachment energies of the anionic clusters as computed with the CASPT2 method are usually in good agreement with the experimental values as observed in the photoelectron spectra.13-15,17-20 To best of our knowledge, the multiconfigurational CASSCF/CASPT2 method has not been employed to study the geometrical and electronic structures of vanadium doped germanium clusters.



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In this work, density functional theory and the multiconfigurational CASSCF/CASPT2 method are utilized to investigate the geometrical and electronic structures of the low-lying states of VGen−/0 (n = 1 - 4) clusters. The CASPT2 method is carried out as the most accurate method to calculate the relative energy order of the low-lying states. Density functional theory with different functionals is also applied to study these clusters. The CASPT2 relative energies are used as a standard to search for the appropriate functionals to study these clusters. Based on the electron detachment energies of the anionic ground states as computed with density functional theory and with the CASPT2 method, we propose new assignments for all features in the photoelectron spectra of VGe3− and VGe4−.

II. Computational Methods Density functional theory (DFT) and multiconfigurational CASSCF/CASPT2 method were applied to investigate the low-lying states of several isomers of VGen−/0 (n = 1 - 4) clusters. The important isomers of these clusters are displayed in Figure 1. The DFT was used for the geometry optimizations and vibrational frequency calculations. In the DFT calculations, the def2-TZVPP basis sets were employed for V and Ge.21 Also, with the purpose to search for the sufficient functional to compute the relative energies of the low-lying states, the BP8622,23, PBE24, B3LYP22,25,26, B3PW9125,27,28, and PBE029,30 functional were applied. The BP86 and PBE are pure GGA functionals, while the B3LYP, B3PW91, and PBE0 are hybrid functions with 20%, 20%, and 25% Hartree-Fock exact exchange. Due to the strong multireference wave functions, the DFT results for the low-lying states of VGe−/0 and VGe2−/0 clusters are not reported. All the DFT calculations were performed with NWCHEM 6.6 package.31


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The CASSCF/CASPT2 was carried out as the most accurate method in this work to investigate the low-lying states of VGen−/0 (n = 1 - 4) clusters. The aug-cc-pVTZ-DK basis sets were applied for V and Ge.32,33 The scalar relativistic effects were included by the second-order Douglas-Kroll Hamiltonian.34-36 The Cholesky decomposition with an accuracy of 10−6 au was employed to reduce the memories for storing two-electron integrals.37-39 In order to prevent the intruder states, an imaginary shift of 0.1 was applied in the CASPT2 calculations. Also, the 3s, 3p of V and 3d of Ge were correlated in the CASPT2 step. In order to get the energies of the excited states, the stage-average CASSCF and multi-state CASPT2 calculations were performed for several roots in each of spatial and spin symmetry. The CASSCF/CASPT2 calculations were performed with MOLCAS@UU 8.0 package.40 The CASSCF was employed to produce the wave functions for the CASPT2 method. The active space orbitals for CASSCF were chosen based on the valence orbitals of V and Ge. For VGe−/0 clusters, the active space included the 3d, 4s of V and 4s, 4p of Ge. In order to increase the flexibility of the active space, five virtual orbitals were added. The resulting active space had 10 or 9 electrons distributed in 15 orbitals that could be labeled as (10,9/15). For VGe2−/0 clusters, two active spaces were employed. The first active space was (10,9/12) which included the 3d, 4s of V and 4p of Ge. This active space was used for the geometry optimization. In order to improve the energy, five virtual orbitals were added to form the (10,9/17) active space which was used in the single-point calculations. All the CASPT2 relative energies of the VGe2−/0 were reported with the larger active space. For the VGe3−/0 and VGe4−/0 clusters, the single-point CASSCF/CASPT2 calculations were performed with the (12,11/14) and (14,13/14) active spaces based on the geometries optimized by the BP86 functional.



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III. Results and Discussion A. VGe−/0 The potential energy profiles of the low-lying states of VGe−/0 clusters as computed with the CASPT2 method are presented in Figure 2a and 2b. Based on the potential energy profiles, the bond distances, vibrational frequencies, and relative energies of the low-lying states are observed and reported in Table 1. These results show that the ground state of the anionic cluster is the 3A1 in C2v symmetry. In C∞v symmetry, this triplet state is labeled as 3Δ. This anionic ground state has a bond distance of 2.310 Å and a vibrational frequency of 301 cm−1. The 5B1 (5Π), 5A1 (5Σ), and 3B1 (3Φ) are less stable than the anionic ground state by 0.35, 0.51, and 0.74 eV, respectively. For the neutral cluster, the 4B1 (4Φ) is the ground state with a bond distance of 2.347 Å and a vibrational frequency of 315 cm−1. The 6A1 (6Σ) is almost degenerate to the ground state with relative energy of 0.04 eV. The 2A1 (2Δ), 2B1 (2Φ), 4A2 (4Δ), 4A1 (4Σ), and 6B1 (6Φ) are less stable than the neutral ground state by 0.31, 0.31, 0.41, 0.45, and 0.72 eV, respectively. It should be noted that in the previous PBE functional calculations, the sextet is reported as the neutral ground state, while the quartet is not investigated.1 The V-Ge distance of the 6A1 (6Σ) as computed with the CASPT2 method is 2.348 Å which is much shorter than that of the sextet state (2.526 Å) as calculated with the PBE functional.1 The molecular orbitals and electron occupation numbers of the 3A1 (3Δ) as calculated with the CASSCF method are displayed in Figure 3. This figure proposes a leading configuration of 13a1214a1215a1116a116b127b106b227b202a20 or 12σ213σ214σ16π47π02δ1 for the anionic ground state. In this configuration, the 13a1 (12σ), 14a1 (13σ), and 6b1, 6b2 (6π) are predominantly 4s, 4p of Ge, while the 15a1, 2a2 (2δ), 16a1 (14σ), 7b1, 7b2 (7π), and 17a1 (15σ) are mainly 3d, 4s of V. This configuration has a reference weight of 51% that implies a strong multireference wave function. Also, the reference 6 ACS Paragon Plus Environment

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weights of the leading configurations of the low-lying states of VGe−/0 clusters as collected in Table 1 are low. The reference weight of the leading configuration of 6A1 (6Σ) is 62%, while that of all the remaining states are much lower. The adiabatic detachment energy (ADE) of the anionic cluster as computed for the 3A1 (3Δ) → 4

B1 (4Φ) transition has a value of 1.48 eV. In this transition, one electron is detached from the bonding

6b1 (6π) orbital. As a result, the V-Ge bond distance increases from 2.363 to 2.409 Å. It can be seen that the CASPT2 ADE of the anionic cluster (1.48 eV) completely differs from the PBE vertical electron affinity of the neutral cluster (0.61 eV).1 This result can be explained by the fact that in the previous PBE calculations the vertical electron affinity of the neutral cluster is computed based on the transition from the excited sextet to the excited quintet state.

B. VGe2−/0 The VGe2−/0 clusters have two important isomers which are named as the cyclic- and the linear-VGe2−/0. Because the preliminary DFT calculations show that the linear isomer is much less stable than the cyclic one, only the latter isomer is investigated with the CASPT2 method. In the cyclic isomer, the Ge2 ligand side-on bonds to the V atom in a C2v symmetry as presented in Figure 1. The leading configurations, V-Ge and Ge-Ge bond distances, and relative energies of the low-lying states of the cyclic-VGe2−/0 isomers as computed with the CASPT2 method are collected in Table 2. The results show that the anionic ground state is the 3A2. The V-Ge and Ge-Ge distances of the ground state are calculated to be 2.351 and 2.497 Å. The 5B1, 3B1, 5A1, 5A2, and 3B2 are above the anionic ground state by 0.18, 0.24, 0.54, 0.56, and 0.58 eV, respectively. For the neutral cluster, it is clear that the ground state is the 4B1, while the 2A1, 4A1, and 2A2 are 0.54, 0.66, and 0.69 eV less stable. The V-Ge 7 ACS Paragon Plus Environment


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and Ge-Ge bond distances of the 4B1 neutral ground state are calculated to be 2.373 and 2.351 Å. These bond distances are respectively shorter than the values of 2.536 and 2.405 Å as computed with the PBE functional.1 The cyclic neutral isomer is also predicted to be the ground state structure of VGe2 cluster by the PBE functional in the previous work although the spin state is not reported.1 The ADE of the 3A2 → 4B1 transition is 1.20 eV as computed with the CASPT2 method. This calculated ADE is 0.35 eV higher than the PBE vertical electron affinity of the neutral cluster (0.85 eV).1 The molecular orbitals and electron occupation numbers of the 3A2 of cyclic-VGe2− isomer are displayed in Figure 4. As can be seen from the figure, the 17a1, 18a1, 7b1, 14b2, 16b2, and 6a2 are predominantly 4p of Ge, while the 19a1, 20a1, 21a1, 8b1, 15b2, and 5a2 are 3d, 4s of V. From this figure,

the

leading

configuration

of

the

3

A2

is

defined

to

be

17a1218a1219a1120a10

7b128b1014b2215b205a21. The 5B1 can be obtained from the 3A2 by transferring one electron from the 14b2 to 20a1 orbital. The 4B1 can be created from the 3A2 by the detachment of one electron from the 14b2 orbital. Because the 14b2 is a bonding orbital between the 3d of V and the π* of Ge2 ligand, the V-Ge distances increase from 2.351 to 2.373 Å, while the Ge-Ge distance decreases from 2.497 to 2.351 Å in the 3A2 → 4B1 transition.

C. VGe3−/0 The low-lying states of tetrahedral-VGe3−/0, cyclic-VGe3−/0, and rhombic-VGe3−/0 isomers as presented in Figure 1 are investigated with the BP86, PBE, B3LYP, B3PW91, and PBE0 functionals and the CASPT2 method. The tetrahedral-VGe3−/0 isomers can have C3v and Cs symmetry, while the cyclicVGe3−/0 and rhombic-VGe3−/0 isomers have C2v symmetry. For computational convenience, Cs symmetry is used for the tetrahedral-VGe3−/0 isomers. The leading configurations and relative 8 ACS Paragon Plus Environment

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energies of the low-lying states of these isomers are collected in Table 3. The CASPT2 results show that the anionic ground state is the 1A′ of tetrahedral-VGe3− isomer. The 3A′ of the same isomer is above the anionic ground state by 0.63 eV. The 1A1 of cyclic-VGe3− and 5A1 of rhombic-VGe3− isomer are respectively 0.89 and 1.10 eV less stable. For the neutral cluster, the 12A′ and 12A″ of tetrahedralVGe3 are the ground states, while the 4B1 of cyclic-VGe3 and 4B2 of rhombic isomer are 0.99 and 0.62 eV less stable. It should be noted that the 12A′ and 12A″ of tetrahedral-VGe3 isomer are formed from the Jahn-Teller distortion of the 12E state in C3v symmetry. The pure GGA and the hybrid functionals predict different spin states as the anionic and neutral ground state. For most of the states, the relative energy order of the BP86 and PBE functional are similar to that of the CASPT2 method although the former is somewhat lower than the latter. In particular, the pure GGA BP86 and PBE functionals calculate the low-spin 1A′ and 12A′ (12A″) states of tetrahedral-VGe3−/0 isomers as the anionic and neutral ground states. In contrast, the hybrid B3LYP, B3PW91, and PBE0 functionals compute the high-spin 5A1 and 4B2 states of rhombic-VGe−/0 isomers to be the ground states. The low-spin 1A′ and 12A′ states of tetrahedral-VGe3−/0 are computed to be 0.26 (0.42, 0.61) and 0.14 (0.19, 0.25) eV less stable than the anionic and neutral ground states by the B3LYP (B3PW91, PBE0) functional. Also, this feature can be seen in the case of the 3A′, 3A″, 4A″, and 4

A′ of the tetrahedral-VGe3−/0 isomers in which the B3LYP, B3PW91, and PBE0 relative energies are

much lower than the BP86 and PBE. All of these results imply that the hybrid functionals overestimate the energies of the high-spin states. The failure of the hybrid functionals in the determination of relative energies of VGe3−/0 clusters can be explained by the contribution of the Hartree-Fock exact exchange in the functionals. The hybrid B3LYP, B3PW91, and PBE0 functionals contain 20, 20, and 25% Hartree-Fock exact exchange. The Hartree-Fock exact exchange accounts for the exchange interactions between 9 ACS Paragon Plus Environment


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electrons with parallel spin because of the Pauli principle. That is the reason why the Hartree-Fock exact exchange and therefore the hybrid B3LYP, B3PW91, and PBE0 functionals tend to favor the high-spin states with more electrons in the same spin. It should be noted that the Hartree-Fock exact exchange of the PBE0 functional is larger than that of the B3LYP and B3PW91 functionals by 5%. Therefore, the failure in relative energies of the low-lying states of the BPE0 is larger than that of the B3LYP and B3PW91 functionals. Indeed, as can be seen in Table 3, in the case of the 5A1 and 4B2 of rhombic-VGe3−/0 isomers, the PBE0 relative energies are more negative than the B3LYP and B3PW91. Overall, based on the DFT and CASPT2 results, it can be said that the pure GGA BP86 and PBE functionals are sufficient to calculate the relative energies of the low-lying states of VGe3−/0 clusters, while the hybrid B3LYP, B3PW91, and PBE0 functionals overestimate the energies of the high-spin states. In the previous B3PW91 calculations, the high-spin quintet and quartet states of rhombicVGe3−/0 isomers are computed to be the ground states of the anionic and neutral clusters.2 In the same work, the first band in the 266 nm photoelectron spectrum of the anionic cluster is assigned to the transition from the quintet to the quartet within the rhombic-VGe3−/0 isomers. The calculated ADE of 1.93 eV is in agreement with the starting point of the first band at 1.73 eV in the photoelectron spectrum. On the contrary, the PBE functional calculates the tetrahedral-VGe3 as the most stable isomer of the neutral cluster.1 The vertical electron affinity of the tetrahedral-VGe3 isomer as computed with the PBE functional is 0.93 eV that is too small as compared to the starting point of the first band at 1.73 eV in the photoelectron spectrum of the anionic cluster. Furthermore, all the remaining bands centered at 2.80, 3.13, 3.40, and 3.60 eV are not interpreted. Because the 1A′ of tetrahedral-VGe3− isomer is the anionic ground state as computed with the pure GGA BP86 and PBE functionals and the CASPT2 method, all features of the photoelectron 10 ACS Paragon Plus Environment

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spectrum of the anionic cluster should be assigned to the electron detachments from this singlet state. The molecular orbitals and electron occupation numbers of the 1A′ of tetrahedral-VGe3− isomer as presented in Figure 5 show a leading configuration of 34a′235a′236a′237a′222a″223a″2 with a reference weight of 69%. The low reference weight of the leading configuration of 1A′ implies that this state has strong multi-reference character. The leading configurations of the low-lying states of tetrahedral-VGe3−/0, cyclic-VGe3−/0, and rhombic-VGe3−/0 isomers as presented in Table 3 show that most of the states have strong multi-reference wave functions. According to the electron selection rules for the photoelectron spectroscopy, only the one-electron detachments are allowed. However, in case the anionic ground state has a multi-reference wave function, both of the one- and twoelectron detachments can be observed in the photoelectron spectrum.14,41-44 It is indeed the case of VGe3− where the wave function of the 1A′ anionic ground state has a reference weight of 69%. The photoelectron spectrum of VGe3− cluster is explained by the one- and two-electron detachments from the 1A′ of tetrahedral-VGe3− isomer. Based on the CASPT2 results, the first band at 2.02 eV is explained by the transitions from the 1A′ to the 12A′ and 12A″ (12E in C3v symmetry). The ADEs of the transitions to 12A′ (12A″) as computed with the CASPT2 method and the pure GGA BP86 and PBE functionals are 1.87 (1.87), 1.84 (1.85), and 1.69 (1.70) eV which correspond well with the experimental value of 1.73 eV. The ADEs of the transitions to 12A′ and 12A″ as computed with BP86 and PBE functionals and the CASPT2 method are much higher than the vertical electron affinity of 0.93 eV of the neutral cluster as calculated with the PBE functional.1 The vertical detachment energy (VDE) of the transition to 12A′ (12A″) as calculated with the CASPT2 method is 1.96 eV which is in agreement with the experimental value of 2.02 eV. In the transitions to the 12A′ and 12A″, one electron is respectively removed from the 37a′ and 23a″ orbital. As can be seen in Figure 5, the 37a′ and 23a″ are mainly bonding orbitals between vanadium atom and the Ge3 ligand. Therefore, the 11 ACS Paragon Plus Environment


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detachment of one electron from 37a′ and 23a″ would result in the increase of V-Ge bond distances. Indeed, the BP86 structural parameters as displayed in Figure 6 show that one V-Ge bond distance increases from 2.337 to 2.423 Å in the transition to 12A′ and two V-Ge bonds elongate from 2.337 to 2.390 Å in the transition to the 12A″. Because of the large difference between the geometries of the initial 1A′ and the final 12A′ and 12A″ states, extensively vibrational progressions are expected to appear in the first band at 2.02 eV. However, these vibrational progressions cannot be resolved in the 266 nm photoelectron spectrum because the vibrational frequencies of the final 12A′ and 12A″ states are small. As computed at the BP86 level, the vibrational frequencies of the 12A′ are 88, 121, 182, 232, 263, and 351 cm−1, while that of the 12A″ are 169, 181, 226, 281, 307, and 377 cm−1. Because of such small vibrational frequencies, the first band is observed in the spectrum as an unresolved broad band.2 The second band centered at 2.80 eV is assigned to the transitions to the 22A′ and 22A″ (22E in C3v symmetry) with VDEs of 2.86 and 2.85 eV. In the transitions to 22A′ and 22A″, one electron is detached from the 36a′ and 22a″ orbital of the anionic ground state. The third band at 3.13 eV is the result of the transitions to 42A′ with VDE of 3.13 eV. The transition to 42A′ is done by the detachment of one electron from the 35a′ orbital. The fourth band at 3.40 eV is attributed to the transitions to 52A′ and 62A′ with VDEs 3.34 and 3.42 eV. The transitions to 52A′ and 62A′ are two-electron detachment processes in which one electron is detached from the 37a′ orbital while another electron is transferred from 37a′ to 38a′ orbital and from 23a″ to 24a″ orbital. The last band at 3.60 eV is attributed to the transitions to 82A′ in which one electron is eliminated from the 34a′ orbital. The VDE of the transition to 82A′ is computed to be 3.68 eV which is in good agreement with the experimental value of 3.60 eV. Overall, all features in the photoelectron spectrum are explained by the electron detachment from the 1A′ ground state of tetrahedral-VGe3− isomer. 12 ACS Paragon Plus Environment

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D. VGe4−/0 The low-lying states of A-VGe4−/0 and B-VGe4−/0 isomers are investigated with the DFT and CASPT2 methods. As displayed in Figure 1, the A-VGe4−/0 and B-VGe4−/0 isomers have bipyramidal shapes in which the V atom respectively locates in the equatorial and axial corner. The computational results as presented in Table 4 show that the hybrid B3LYP, B3PW91, and PBE0 functionals again overestimate the energies of the high-spin states as compared to the pure GGA BP86 and PBE functionals and the CASPT2 method. For the anionic cluster, the low-spin 1A′ (Cs) of A-VGe4− isomer is predicted to be the ground state by the BP86 and PBE functionals and the CASPT2 method. It should be noted that the 1A′ is formed from the distortion of the 1A1 in C2v symmetry because of an imaginary vibrational frequency of 113i cm−1. This kind of distortion can be seen in several low-lying states of AVGe4−/0 and B-VGe4−/0 isomers as presented in Table 4. The high-spin 3A′ of A-VGe4− isomer and 3A′, 3

A″, and 5A″ states of B-V-Ge4− isomer are 0.47, 0.55, and 0.59 eV less stable than the ground state as

computed at the CASPT2 level. The BP86 and PBE functionals also give the almost the same stability for these states although the BP86 and PBE relative energies are slightly lower than the CASPT2. In contrast to the BP86 and PBE functionals and the CASPT2 method, the hybrid B3LYP, B3PW91, and PBE0 functionals predict the high-spin 3A′ of the A-VGe4− isomer as the anionic ground state, while the 1

A′ is 0.25, 0.41, and 0.59 eV less stable. For the neutral cluster, the BP86 and PBE functionals predict

the 12A′ (Cs) of A-VGe4 isomer to be the ground state. The 12A′ is the result of the distortion of the 2A1 (C2v) because of an imaginary frequency of 73i cm−1 as computed at the BP86 level. At the BP86 and PBE level, the energy relaxation of this distortion is only 0.03 and 0.02 eV. The CASPT2 method favors the higher symmetry 2A1 than the lower symmetry 12A′ by stabilizing the initial state by only 0.05 eV. 13 ACS Paragon Plus Environment


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The tiny energy difference between the 12A′ (Cs) and the 2A1 (C2v) implies that the A-VGe4 isomer can exist in Cs and C2v symmetries. On the other hand, at the B3LYP, B3PW91, and PBE0 levels, the 6A1 of A-VGe4 is almost degenerate with the 12A′ with relative energies of 0.03, −0.10, and −0.14 eV. However, at the BP86, PBE, and CASPT2 levels, this sextet state is 0.50, 0.44, and 0.57 eV less stable than the 12A′. Because the CASPT2 is carried out as the most believable method in this work, it is suggested that the ground states of VGe4−/0 clusters are the 1A′ and 12A′ of A-VGe4−/0 isomers. The previous DFT calculations with B3PW91 functional show that the ground state of the anionic cluster is a triplet state of A-VGe4− isomer in C2v symmetry.2 However, the DFT and CASPT2 results in this work propose that the anionic ground state is not the triplet but the 1A′ of A-VGe4− isomer in Cs symmetry. Therefore, the 266 nm photoelectron spectrum of VGe4− cluster should be interpreted based on the one-electron detachments from the 1A′ anionic ground state. The photoelectron spectrum of VGe4− has five bands centered at 2.47, 2.84, 3.20, 3.60, and 4.09 eV.2 The electron detachment energies as computed with DFT and CASPT2 method are collected in Table 4. The molecular orbitals and electron occupation numbers of the 1A′ of A-VGe4− isomer as displayed in Figure 7 propose a leading configuration of 45a′246a′247a′248a′226a″227a″228a″2. Starting from the 1

A′, the first band at 2.47 eV is the result of the transition to 12A′ and 12A″ with CASPT2 VDEs of 2.49

and 2.53 eV. In these two transitions, one electron is detached from the 48a′ and 28a″ orbitals. The ADEs of the transition to 12A′ as calculated with the BP86 and PBE functionals and the CASPT2 method are 2.10, 1.94, and 2.29 eV which are in agreement with the experimental value of 2.20 eV. These computed ADEs are much higher than the vertical electron affinity of 1.515 eV of the neutral cluster as calculated with the PBE functional.1 The second and third bands at 2.84 and 3.20 eV are ascribed to the transitions to 22A′ and 22A″ with CASPT2 VDE of 2.95 and 3.23 eV. In the transition to 22A′ and 22A″, one electron is respectively removed from the 47a′ and 27a″ orbitals. The fourth band at 3.60 14 ACS Paragon Plus Environment

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eV are attributed to the transitions to the 42A′ and 42A″ with CASPT2 VDEs of 3.76 and 3.79 eV. In the transition to 32A′ and 42A″, one electron is respectively detached from the 46a′ and 26a″ orbitals. The last band at 4.06 eV is explained by the transition from 1A′ to the 62A′ in which one electron is detached from the 45a′ orbital. The VDE of this transition is computed to be 3.91 eV by the CASPT2 method.

IV. Conclusion The low-lying states of VGen−/0 (n = 1 - 4) clusters are investigated with the DFT and CASPT2 methods. From the computational results, the most stable isomers of the VGen−/0 (n = 1 - 4) are predicted to be the linear-VGe−/0, cyclic-VGe2-/0, tetrahedral-VGe3−/0, and bipyramidal A-VGe4−/0 isomers. In these isomers, vanadium atom tends to interact with more Ge atoms. For the VGe−/0 clusters, the CASPT2 results propose the 3A1 (3Δ) and 4B1 (4Φ) as the anionic and neutral ground states. The ADE of the transition from the anionic to the neutral ground state is evaluated to be 1.48 eV. For the VGe2−/0 clusters, the 3A2 and 4B1 of cyclic-VGe2−/0 isomers are computed to be the anionic and neutral ground states by the CASPT2 method. The ADE of the 3A2 of cyclic-VGe2−/0 isomer is computed to be 1.20 eV. For the VGe3−/0 and VGe4−/0 clusters, the computational results show that the hybrid B3LYP, B3PW91, and PBE0 functionals overestimated the energies of the high-spin states, while the pure GGA BP86 and PBE functionals give almost the same relative energy order for the lowlying states as compared to the CASPT2 method. The failure of the hybrid functionals is explained by the large contribution of the Hartree-Fock exact exchange. Based on the pure GGA BP86 and PBE functional and the CASPT2 results, the ground states of VGe3−/0 clusters are predicted to be the 1A′ and 12A′ (12A″) of the tetrahedral-VGe3−/0 isomers, while that of the VGe4−/0 clusters are calculated to 15 ACS Paragon Plus Environment


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be 1A′ and 12A′ of the A-VGe4−/0 isomers. The ADEs and VDEs of the ground state of VGe3− and VGe4− clusters are computed with the BP86 and PBE functionals and the CASPT2 method. Based on the electron detachments from the anionic ground states, all the important features in the photoelectron spectra of VGe3− and VGe4− are explained.

Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 104.06-2016.16.

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(28) Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Erratum: Atoms, Molecules, Solids, and Surfaces: Applications of the Generalized Gradient Approximation for Exchange and Correlation. Phys. Rev. B 1993, 48, 4978-4978. (29) Adamo, C.; Scuseria, G. E.; Barone, V. Accurate Excitation Energies from Time-Dependent Density Functional Theory: Assessing the PBE0 Model. J. Chem. Phys. 1999, 111, 2889-2899. (30) Adamo, C.; Barone, V. Toward Reliable Density Functional Methods without Adjustable Parameters: The PBE0 Model. J. Chem. Phys. 1999, 110, 6158-6170. (31) Valiev, M.; Bylaska, E. J.; Govind, N.; Kowalski, K.; Straatsma, T. P.; Van Dam, H. J. J.; Wang, D.; Nieplocha, J.; Apra, E.; Windus, T. L.; et al. NWChem: A Comprehensive and Scalable Open-Source Solution for Large Scale Molecular Simulations. Comput. Phys. Commun. 2010, 181, 1477-1489. (32) Balabanov, N. B.; Peterson, K. A. Systematically Convergent Basis Sets for Transition Metals. I. AllElectron Correlation Consistent Basis Sets for the 3d Elements Sc-Zn. J. Chem. Phys. 2005, 123, 064107. (33) Wilson, A. K.; Woon, D. E.; Peterson, K. A.; Dunning, T. H. Gaussian Basis Sets for Use in Correlated Molecular Calculations. IX. The Atoms Gallium through Krypton. J. Chem. Phys. 1999, 110, 7667-7676. (34) Ishikawa, Y.; Vilkas, M. J. Relativistic Quantum Mechanics of Many-Electron Systems. J. Mol. Struc. THEOCHEM 2001, 573, 139-169. (35) Reiher, M.; Wolf, A. Exact Decoupling of the Dirac Hamiltonian. II. The Generalized Douglas-Kroll-Hess Transformation up to Arbitrary Order. J. Chem. Phys. 2004, 121, 10945-10956. (36) Reiher, M.; Wolf, A. Exact Decoupling of the Dirac Hamiltonian. I. General Theory. J. Chem. Phys. 2004, 121, 2037-2047. (37) Aquilante, F.; Lindh, R.; Bondo Pedersen, T. Unbiased Auxiliary Basis Sets for Accurate Two-Electron Integral Approximations. J. Chem. Phys. 2007, 127, 114107. (38) Aquilante, F.; Malmqvist, P.-Å.; Pedersen, T. B.; Ghosh, A.; Roos, B. O. Cholesky Decomposition-Based Multiconfiguration Second-Order Perturbation Theory (CD-CASPT2): Application to the Spin-State Energetics of CoIII(diiminato)(NPh). J. Chem. Theory Comput. 2008, 4, 694-702. (39) Aquilante, F.; Pedersen, T. B.; Lindh, R.; Roos, B. O.; Sánchez de Merás, A.; Koch, H. Accurate Ab Initio Density Fitting for Multiconfigurational Self-Consistent Field Methods. J. Chem. Phys. 2008, 129, 024113. (40) Aquilante, F.; Autschbach, J.; Carlson, R. K.; Chibotaru, L. F.; Delcey, M. G.; De Vico, L.; Fdez. Galván, I.; Ferré, N.; Frutos, L. M.; Gagliardi, L.; et al. Molcas 8: New Capabilities for Multiconfigurational Quantum Chemical Calculations Across the Periodic Table. J. Comput. Chem. 2016, 37, 506-541. (41) Li, W.-L.; Su, J.; Jian, T.; Lopez, G. V.; Hu, H.-S.; Cao, G.-J.; Li, J.; Wang, L.-S. Strong Electron Correlation − in UO2 : A Photoelectron Spectroscopy and Relativistic Quantum Chemistry Study. J. Chem. Phys. 2014, 140, 094306. (42) Wu, H.; Desai, S. R.; Wang, L.-S. Chemical Bonding between Cu and Oxygen - Copper Oxides vs O2 Complexes: A Study of CuOx (x = 0-6) Species by Anion Photoelectron Spectroscopy. J. Phys. Chem. A 1997, 101, 2103-2111. (43) Wu, H.; Wang, L.-S. Photoelectron Spectroscopy and Electronic Structure of ScOn− (n = 1-4) and YOn− (n = 1-5): Strong Electron Correlation Effects in ScO− and YO−. J. Phys. Chem. A 1998, 102, 9129-9135. (44) Wu, H.; Wang, L.-S. Electronic Structure of Titanium Oxide Clusters: TiOy (y = 1-3) and (TiO2)n (n = 1-4). J. Chem. Phys. 1997, 107, 8221-8228.


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−/0

Table 1. The leading configurations, V-Ge bond distances, vibrational frequencies, and relative energies (REs) of the low-lying states of VGe computed with the CASPT2 method. cluster − VGe

VGe

state 3 A1 ( Δ) 5 5 B1 ( Π) 5 5 A1 ( Σ) 3 3 B1 ( Φ) 4 4 B1 ( Φ) 6 6 A1 ( Σ) 2 2 A1 ( Δ) 2 2 B1 ( Φ) 4 4 A2 ( Δ) 4 4 A1 ( Σ) 6 6 B1 ( Φ) 3

leading configuration 2 1 4 0 1 12σ 13σ 14σ 6π 7π 2δ (51%) 2 2 1 3 0 2 12σ 13σ 14σ 6π 7π 2δ (61%) 2 2 1 3 1 1 12σ 13σ 14σ 6π 7π 2δ (29%) 2 2 2 3 0 1 12σ 13σ 14σ 6π 7π 2δ (40%) 2 2 1 3 0 1 12σ 13σ 14σ 6π 7π 2δ (50%) 2 2 1 2 0 2 12σ 13σ 14σ 6π 7π 2δ (62%) 2 2 0 4 0 1 12σ 13σ 14σ 6π 7π 2δ (39%) 2 2 1 3 0 1 12σ 13σ 14σ 6π 7π 2δ (21%) 2 1 1 4 0 1 12σ 13σ 14σ 6π 7π 2δ (55%) 2 2 1 2 0 2 12σ 13σ 14σ 6π 7π 2δ (35%) 2 2 1 2 1 1 12σ 13σ 14σ 6π 7π 2δ (48%) 2

3

V-Ge (Å) 2.310 2.370 2.411 2.395 2.347 2.348 2.340 2.380 2.249 2.459 2.494

−1

frequency (cm ) 301 282 360 301 315 406 284 346 349 269 230

clusters as

(a)

RE (eV) 0.00 0.35 0.51 0.74 0.00 (1.48) 0.04 0.31 0.31 0.41 0.45 0.72

3

(a) The number in parentheses are adiabatic detachment energy of the A1 ( Δ) anion ground state.



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−/0

Table 2. The leading configurations, V-Ge and Ge-Ge bond distances, and relative energies (REs) of the low-lying states of the cyclic VGe2 isomer − cyclic-VGe2

cyclic-VGe2

state 3 A2 5 B1 3 B1 5 A1 5 A2 3 B2 4 B1 2 A1 4 A1 2 A2

leading configuration 2 1 0 2 0 2 0 1 18a1 19a1 20a1 7b1 8b1 14b2 15b2 5a2 (64%) 2 1 1 2 0 1 0 1 18a1 19a1 20a1 7b1 8b1 14b2 15b2 5a2 (81%) 2 2 0 2 0 1 0 1 18a1 19a1 20a1 7b1 8b1 14b2 15b2 5a2 (33%) 2 1 0 2 1 1 0 1 18a1 19a1 20a1 7b1 8b1 14b2 15b2 5a2 (84%) 2 1 0 2 0 1 1 1 18a1 19a1 20a1 7b1 8b1 14b2 15b2 5a2 (76%) 2 0 0 2 1 2 0 1 18a1 19a1 20a1 7b1 8b1 14b2 15b2 5a2 (64%) 2 1 0 2 0 1 0 1 18a1 19a1 20a1 7b1 8b1 14b2 15b2 5a2 (77%) 2 1 0 2 0 2 0 0 18a1 19a1 20a1 7b1 8b1 14b2 15b2 5a2 (65%) 2 0 0 2 1 1 0 1 18a1 19a1 20a1 7b1 8b1 14b2 15b2 5a2 (80%) 2 0 0 2 0 2 0 1 18a1 19a1 20a1 7b1 8b1 14b2 15b2 5a2 (69%)

R (V-Ge, Ge-Ge) (Å) 2.351, 2.497 2.389, 2.369 2.388, 2.333 2.433, 2.348 2.369, 2.486 2.366, 2.498 2.373, 2.351 2.370, 2.424 2.410, 2.326 2.341, 2.452

isomers.

(a)

RE (eV) 0.00 0.18 0.24 0.54 0.56 0.58 0.00 (1.20) 0.54 0.66 0.69

3

(a) The number in parentheses is adiabatic detachment energy of the A2 anionic ground state.


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−/0

Table 3. The relative energies of the low-lying states of VGe3

clusters as computed with the density functional theory and CASPT2 method. (a)

isomer − tetrahetral-VGe3 −

cyclic-VGe3

−

rhombic-VGe3 tetrahetral-VGe3

cyclic-VGe3 rhombic-VGe3

state 1 A′ 3 A′ 3 A″ 1 A1 3 A2 3 1 B2 5 A1 2 1 A′ 2 1 A″ 2 2 A′ 2 2 A″ 2 3 A′ 2 4 A′ 2 5 A′ 2 6 A′ 2 7 A′ 2 8 A′ 4 A″ 4 A′ 4 B1 6 A1 4 B2

leading configurartion 2 2 2 2 0 0 2 2 0 34a′ 35a′ 36a′ 37a′ 38a′ 39a′ 22a″ 23a″ 24a″ (69%) 2 2 2 1 1 0 2 2 0 34a′ 35a′ 36a′ 37a′ 38a′ 39a′ 22a″ 23a″ 24a″ (51%) 2 2 2 1 0 0 2 2 1 34a′ 35a′ 36a′ 37a′ 38a′ 39a′ 22a″ 23a″ 24a″ (34%) 2 2 0 2 0 2 2 0 2 25a1 26a1 27a1 10b1 11b1 17b2 18b2 19b2 6a2 (65%) 2 2 0 2 1 2 1 0 2 25a1 26a1 27a1 10b1 11b1 17b2 18b2 19b2 6a2 (59%) 2 2 1 2 0 2 1 0 2 25a1 26a1 27a1 10b1 11b1 17b2 18b2 19b2 6a2 (35%) 2 2 1 2 1 2 1 1 25a1 26a1 27a1 10b1 11b1 17b2 18b2 6a2 (48%) 2 2 2 1 0 0 2 2 0 34a′ 35a′ 36a′ 37a′ 38a′ 39a′ 22a″ 23a″ 24a″ (67%) 2 2 2 2 0 0 2 1 0 34a′ 35a′ 36a′ 37a′ 38a′ 39a′ 22a″ 23a″ 24a″ (67%) 2 2 1 2 0 0 2 2 0 34a′ 35a′ 36a′ 37a′ 38a′ 39a′ 22a″ 23a″ 24a″ (65%) 2 2 2 2 0 0 1 2 0 34a′ 35a′ 36a′ 37a′ 38a′ 39a′ 22a″ 23a″ 24a″ (67%) 2 2 2 2 1 0 2 0 0 34a′ 35a′ 36a′ 37a′ 38a′ 39a′ 22a″ 23a″ 24a″ (27%) 2 1 2 2 0 0 2 2 0 34a′ 35a′ 36a′ 37a′ 38a′ 39a′ 22a″ 23a″ 24a″ (34%) 2 2 2 0 1 0 2 2 0 34a′ 35a′ 36a′ 37a′ 38a′ 39a′ 22a″ 23a″ 24a″ (24%) 2 2 2 1 0 0 2 1 1 34a′ 35a′ 36a′ 37a′ 38a′ 39a′ 22a″ 23a″ 24a″ (13%) 2 2 2 1 0 0 2 1 1 34a′ 35a′ 36a′ 37a′ 38a′ 39a′ 22a″ 23a″ 24a″ (15%) 1 2 2 2 0 0 2 2 0 34a′ 35a′ 36a′ 37a′ 38a′ 39a′ 22a″ 23a″ 24a″ (43%) 2 2 2 1 1 0 2 1 0 34a′ 35a′ 36a′ 37a′ 38a′ 39a′ 22a″ 23a″ 24a″ (41%) 2 2 2 1 0 0 2 1 1 34a′ 35a′ 36a′ 37a′ 38a′ 39a′ 22a″ 23a″ 24a″ (60%) 2 2 1 2 0 2 1 0 1 25a1 26a1 27a1 10b1 11b1 17b2 18b2 19b2 6a2 (58%) 2 1 1 2 1 2 1 0 1 25a1 26a1 27a1 10b1 11b1 17b2 18b2 19b2 6a2 (63%) 2 2 1 2 1 2 0 1 25a1 26a1 27a1 10b1 11b1 17b2 18b2 6a2 (59%)

BP86 0.00 0.45 0.73 0.82 0.95 0.59 0.55 0.00 (1.84) 0.01 (1.85)

0.30 0.56 0.73 0.76 0.36

relative energy (eV) PBE B3LYP B3PW91 0.00 0.00 0.00 0.41 0.09 −0.04 0.71 0.15 0.19 0.83 0.49 0.40 0.95 0.43 0.38 0.59 0.24 0.16 0.51 −0.26 −0.42 0.00 (1.69) 0.00 0.00 0.01 (1.70) 0.01 0.01

0.25 0.53 0.74 0.75 0.34

1

−0.08 0.17 0.31 0.30 −0.14

−0.14 0.11 0.37 0.21 −0.19

PBE0 0.00 −0.19 0.02 0.23 0.22 0.02 −0.61 0.00 0.01

−0.19 0.05 0.34 0.19 −0.25

CASPT2 0.00 0.63 0.80 0.89 1.00 0.75 1.10 0.00 (1.87) 0.00 (1.87)

VDE (eV) (b) CASPT2 expt.

1.96 1.96 2.86 2.85 3.09 3.13 3.34 3.42 3.61 3.68

2.02 2.02 2.80 2.80 3.13 3.40 3.40 3.60

0.31 0.63 0.99 1.23 0.62

−

(a) The numbers in parentheses are adiabatic and vertical detachment energies of the A′ of tetrahedral-VGe3 isomer. 2

(b) Reference .



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−/0

Table 4. The relative energies of the low-lying states of VGe4

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clusters as computed with the density functional theory and CASPT2 method. (b)

isomer − A-VGe4

−

B-VGe4

A-VGe4

B-VGe4

state 1 A′ 1 A1 3 A′ 3 A1 3 A2 3 A′ 3 A″ 5 A″ 2 1 A′ 2 1 A″ 2 2 A′ 2 2 A″ 2 3 A′ 2 3 A″ 2 4 A′ 2 4 A″ 2 5 A′ 2 6 A′ 2 A1 2 B2 4 B1 4 B2 6 A1 2 A 2 A′ 2 A″ 4 A″

symmetry Cs C2v Cs C2v C2v Cs Cs Cs Cs Cs Cs Cs Cs Cs Cs Cs Cs Cs C2v C2v C2v C2v C2v C1 Cs Cs Cs

−1 (a)

frequency (cm ) 93, 100, 123, 142, 203, 218, 237, 320, 335 113i, 93, 149, 150, 184, 204, 206, 320, 324 63, 114, 119, 143, 191, 196, 213, 263, 286 59i, 113, 138, 156, 181, 183, 200, 259, 280 93, 109, 133, 146, 158, 184, 208, 250, 274 70, 87, 125, 147, 165, 181, 205, 240, 274 74, 94, 146, 151, 159, 190, 200, 239, 273 71, 102, 113, 141, 180, 184, 216, 227, 560 73, 119, 128, 133, 197, 208, 223, 264, 307 142, 161, 164, 185, 221, 235, 273, 345, 356

73i, 117, 147, 163, 180, 189, 208, 257, 295 50i, 52, 90, 116, 154, 175, 209, 251, 326 149, 153, 190, 206, 207, 219, 236, 275, 288 146, 152, 179, 193, 209, 224, 225, 291, 293 77, 87, 118, 143, 161, 173, 190, 237, 253 82, 107, 141, 150, 168, 203, 216, 240, 303 32i, 88, 96, 129, 170, 180, 182, 238, 300 154, 171, 193, 194, 205, 258, 263, 265, 347 64, 103, 103, 124, 186, 206, 222, 238, 271

BP86 0.00 0.26 0.19 0.21 0.47 0.30 0.30 0.49 0.00 (2.10) 0.17 (2.27)

0.03 0.20 0.36 0.38 0.50 0.19 0.20 0.22 0.28

relative energy (eV) PBE B3LYP B3PW91 0.00 0.00 0.00 0.23 0.26 0.19 0.15 −0.25 −0.41 0.16 −0.25 −0.41 0.42 −0.06 −0.24 0.25 −0.12 −0.26 0.25 −0.14 −0.27 0.45 −0.09 −0.24 0.00 (1.94) 0.00 0.00 0.20 (2.14) 0.39 -

0.02 0.22 0.33 0.36 0.44 0.17 0.17 0.21 0.27

0.00 0.41 0.01 0.06 0.03 0.29 0.31 0.30 0.04

0.00 0.50 −0.06 0.02 −0.10 0.23 0.23 0.27 0.01

PBE0 0.00 0.16 −0.59 −0.59 −0.44 −0.43 −0.45 −0.40 0.00 0.58

0.00 0.67 −0.10 −0.02 −0.14 0.24 0.26 0.30 −0.02

CASPT2 0.00 0.11 0.47 0.42 0.58 0.55 0.59 1.13 0.00 (2.29) 0.18 (2.48)

VDE (eV) (c) CASPT2 expt.

2.49 2.53 2.93 3.23 3.47 3.69 3.76 3.79 3.79 3.91

2.47 2.47 2.84 3.20

3.60 3.60 4.09

−0.05 0.25 0.72 0.82 0.57 0.15 0.10 0.11 0.48

(a) The vibrational frequencies are reported with the BP86 functional. (b) The numbers in parentheses are adiabatic detachment energies. 2

(c) Reference .


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Figure 1. The geometrical structures of the important isomers of VGen

−/0

(n = 1-4) clusters.



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(a)

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(b)

−

Figure 2. The potential energy profiles of the low-lying states of VGe (a) and VGe (b) as computed with the CASPT2 method.


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3

3

−

Figure 3. The natural molecular orbitals and electron occupation numbers of the A1 ( Δ) of VGe cluster as calculated with the CASSCF method.



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Page 26 of 30

3

−

Figure 4. The natural molecular orbitals and electron occupation numbers of the A2 of cyclic-VGe2 isomer as calculated with the CASSCF method.


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1

−

Figure 5. The natural molecular orbitals and electron occupation numbers of the A′ of tetrahedral-VGe3 isomer as calculated with the CASSCF method.



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1

2

1

2

Page 28 of 30

−/0

Figure 6. The structural relaxation in the A′ → 1 A′ and A′ → 1 A″ transitions within the tetrahedral-VGe3 isomers as computed with the BP86 functional.


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1

−

Figure 7. The natural molecular orbitals and electron occupation numbers of the A′ of A-VGe4 isomer as obtained from the CASSCF calculations.



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